CHAPTER 5.1Using Fundamental Identities

USING THE UNIT CIRCLE TO FIND REMAINING TRIGONOMETRIC VALUES When given a trig function, there are two

values of the triangle’s sides that are inherently given to us:

When given two trig functions, it is possible to determine which quadrant/axis contains θ

Using these values and the Pythagorean Thm it is possible to find the remaining side

Using all three sides it is possible to find the remaining trig functions

Using previous notes and the back cover of the book reference sheet for all the formulas to help you through some of these problems.

Hint: The ones you memorized in Chapter 4

The fundamental trigonometric identities come in several related groups:

Reciprocal IdentitiesQuotient IdentitiesCofunction IdentitiesPythagorean IdentitiesEven-Odd Identities

EX 1: GIVEN A CSC Θ AND TAN ΘUSING THE ∆ TO SOLVE FOR THE TRIG F(X)

opp

hyp

sin

1csc

3

5csc

opp

hyp

3

θ

Quad 1

4

tan sincos

opp

adj

tan opp

adj

3

4

EX 1: GIVEN A CSC Θ AND TAN Θ

222 cba 3

θ4

2

2

222

25

169

43

c

c

c

5

EX 1: GIVEN A CSC Θ AND TAN Θ

3

θ4

sin 3

5 5

cos 4

5

sec 5

4

cot 4

3

EX 2: GIVEN A COT Θ AND COS ΘUSING THE UNIT CIRCLE TO SOLVE FOR THE TRIG F(X)

opp

adj

sin

coscot

0

1cot

cot

opp

adj

DNE

hyp

adjcos

0cos

,0

Θ must be in Quad 1, Quad 4 or the positive x-axis

EX 2: GIVEN A COT Θ AND COS Θ

,0 Θ must be in Quad 1, 4 or the positive x-axis

Q1(+, +)

Q2(-, +)

Q3(-, -)

Q4(+, -)

sin θ = +cos θ = +tan θ = +

sin θ = +cos θ = -tan θ = -

sin θ = -cos θ = -tan θ = +

sin θ = -cos θ = +tan θ = -

Θ must be 0

EX 2: GIVEN A COT Θ AND COS Θ

You can’t draw a triangle with θ = 0,

But you do have the unit circle memorized as (1, 0)

which allows you to find the six trig functions.

1cos 1sec

DNEcot

0sin DNEcsc

0tan

PROVING TRIG FUNCTIONS ARE EQUAL

On top of the previous notes, there are some equivalent trig functions

cos2 x 1 sin2 x

sin2 x 1 cos2 x

tan2 x sec2 x 1

cot 2 x csc2 x 1

VERIFYING IDENTITIESVERIFYING IDENTITIES

In order to verify an equation is an identity, you must follow these steps:

1)Start with the expression on one side of the equation (Hint: pick the “harder” side.)

2)Manipulate that expression using known identities (Hint: put everything into sin θ and cos θ)

3)Stop when you reach the expression on the other side of the equation

EX 3: VERIFY THE IDENTITY

cot x sec x sin x = 1

11

sin

cos

1

sin

cosx

xx

x

1 =1

EX 4: VERIFY THE IDENTITY

csc x csc x cos2 x sin x

OOOh, there’s a common factor!

xxx sinsin csc 2

xxx sin)cos1(csc 2

xxx

sinsin sin

1 2

xx sinsin

EX 5: VERIFYING THE IDENTITY

tan cot sec csc

cscsecsin

cos

cos

sin

cscseccos

cos

sin

cos

sin

sin

cos

sin

cscsec

cossin

cos

cossin

sin 22

EX 5: VERIFYING THE IDENTITY

cscsec

cossin

cos

cossin

sin 22

cscsec

cossin

cossin 22

cscseccos

1

sin

1

cscsecseccsc

cscseccscsec